A class for the $M \to P l^+ l^-$ decay. More...

#include <MPll.h>

Detailed Description

A class for the $M \to P l^+ l^-$ decay.

Author: HEPfit Collaboration

Copyright: GNU General Public License

This class is used to compute all the functions needed in order to build the observables relative to the $M \to P l^+ l^-$ decays, where $M$ is a generic meson and $P$ is a pseudoscalar meson.

MPll parameters

The mandatory parameters of MPll are summarized below:

Label	LaTeX symbol	Description
r_1_fplus, r_2_fplus, m_fit2_fplus	$r_1^{f_+}, r_2^{f_+}, m_{fit}^{2,f_+}$	The fit parameters for the LCSR form factor $f_+$ of the $B\to K$.
r_1_fT, r_2_fT, m_fit2_fT	$r_1^{f_T}, r_2^{f_T}, m_{fit}^{2,f_T}$	The fit parameters for the LCSR form factor $f_T$ of the $B\to K$.
r_2_f0, m_fit2_f0	$r_2^{f_0}, m_{fit}^{2,f_0}$	The fit parameters for the LCSR form factor $f_0$ of the $B\to K$.
absh_0_MP, absh_0_1_MP	$\mathrm{Abs}h_0^{(0)}, \mathrm{Abs}h_0^{(1)}$	The constant and linear terms of the absolute value of the hadronic parameter $h_0$ of the $B\to K$.
argh_0_MP, argh_0_1_MP	$\mathrm{Arg}h_0^{(0)}, \mathrm{Arg}h_0^{(1)}$	The constant and linear terms of the argument of the hadronic parameter $h_0$ of the $B\to K$.

This kind of decays can be described by means of the $\Delta B = 1 $ weak effective Hamiltonian

\[ \mathcal{H}_\mathrm{eff}^{\Delta B = 1} = \mathcal{H}_\mathrm{eff}^\mathrm{had} + \mathcal{H}_\mathrm{eff}^\mathrm{sl+\gamma}, \]

where the first term is the hadronic contribution

\[ \mathcal{H}_\mathrm{eff}^\mathrm{had} = \frac{4G_F}{\sqrt{2}}\Bigg[\sum_{p=u,c}\lambda_p\bigg(C_1 Q^{p}_1 + C_2 Q^{p}_2\bigg) -\lambda_t \bigg(\sum_{i=3}^{6} C_i P_i + C_{8}Q_{8g} \bigg)\Bigg] \,, \]

involving current-current, chromodynamic penguin and chromomagnetic dipole operators, while the second one, given by

\[ \mathcal{H}_\mathrm{eff}^\mathrm{sl+\gamma} = - \frac{4G_F}{\sqrt{2}}\lambda_t \bigg( C_7Q_{7\gamma} + C_9Q_{9V} + C_{10}Q_{10A} \bigg) \,, \]

includes the electromagnetic penguin plus the semileptonic operators.

Considering the matrix element of $\mathcal{H}_\mathrm{eff}^{\Delta B = 1}$ between the initial state $M$ and the final state $P l^+ l^-$, only the contribution of $\mathcal{H}_\mathrm{eff}^\mathrm{sl+\gamma}$ clearly factorizes into the product of hadronic form factors and leptonic tensors at all orders in strong interactions. Following [Jager:2012uw], we implemented the amplitude in the helicity basis; hence we made use of the helicity form factors $ \tilde{V}_0(q^2), \tilde{T}_0(q^2)$ and $\tilde{S}(q^2) $, which are related to the ones in the transverse basis through the following relations :

\[ \tilde{V}_0(q^2) = i \frac{\sqrt{\lambda(q^2)}}{2m_M\sqrt{q^2}}f_+(q^2)\,,\\ \tilde{T}_0(q^2) = i \frac{\sqrt{\lambda(q^2)q^2}}{2m_M^2(m_M+m_P)}f_T(q^2)\,,\\ \tilde{S}(q^2) = -\frac{m_M^2-m_P^2}{2m_M(m_b+m_s)}\frac{1+m_s/m_b}{1-m_s/m_b}f_0(q^2)\,, \]

where $\lambda(q^2) = 4m_M^2|\vec{k}|^2$, with $\vec{k}$ as the 3-momentum of the meson $P$ in the $M$ rest frame.

The effect of the operators of $\mathcal{H}_\mathrm{eff}^\mathrm{had}$ due to exchange of soft gluon can be reabsorbed in the following parameterization,

\[ h_0(q^2) = \frac{\epsilon^*_\mu(\lambda)}{m_M^2} \int d^4x e^{iqx} \langle \bar P \vert T\{j^{\mu}_\mathrm{em} (x) \mathcal{H}_\mathrm{eff}^\mathrm{had} (0)\} \vert \bar M \rangle = h_0^{(0)} + \frac{q^2}{1\,\mathrm{GeV}^2} h_0^{(1)}\,. \]

The amplitude can be therefore parametrized in terms of the following helicity amplitudes:

\[ H_V = -i\, N \Big\{C_{9} \tilde{V}_{L,0} +C_{9}' \tilde{V}_{R,0} + \frac{m_M^2}{q^2} \Big[\frac{2\, m_b}{m_M} (C_{7} \tilde{T}_{L,0} + C_{7}' \tilde{T}_{R,0}) - 16 \pi^2 h_0 \Big] \Big\} \,, \\ H_A = -i\, N (C_{10} \tilde{V}_{L,0} + C_{10}'\tilde{V}_{R,0}) \,, \\ H_S = i\, N \frac{ m_b}{m_W} (C_S \tilde{S}_L + C_S' \tilde{S}_R)\,, \\ H_P = i\, N \Big\{ \frac{ m_b}{m_W} (C_P \tilde{S}_L + C_P' \tilde{S}_R) + \frac{2\,m_\ell m_b}{q^2} \left[C_{10} \Big(\tilde{S}_L - \frac{m_s}{m_b} \tilde{S}_R \Big) + C_{10}' \Big(\tilde{S}_R - \frac{m_s}{m_b} \tilde{S}_L\Big) \right] \Big\} \,, \]

where $ N = - \frac{4 G_F m_M}{\sqrt{2}}\frac{e^2}{16\pi^2}\lambda_t$ and we have defined

\[ \tilde{V}_{L,0}(q^2) = -\tilde{V}_{R,0}(q^2)=\tilde{V}_0(q^2)\,,\\ \tilde{T}_{L,0}(q^2) = -\tilde{T}_{R,0}(q^2)=\tilde{V}_0(q^2)\,,\\ \tilde{S}_L(q^2) = -\tilde{S}_R(q^2)=\tilde{S}(q^2)\,. \]

Squaring the amplitude and summing over the spins it is possible to obtain the fully differential decay rate, which is

\[ \frac{d^{(4)} \Gamma}{dq^2\,d(\cos\theta_l)} = \frac{9}{32\,\pi} \Big( I^c_1 +I^c_2\cos2\theta_l + I_6^c \cos\theta_l \Big) \]

The angular coefficients involved in the differential decay rate are related to the helicity amplitudes according to the following relations:

\[ I_1^c = F \left\{ \frac{1}{2}\left(|H_V^0|^2+|H_A^0|^2\right)+ |H_P|^2+\frac{2m_\ell^2}{q^2}\left(|H_V^0|^2-|H_A^0|^2\right) + \beta^2 |H_S|^2 \right\}\,,\\ I_2^c = -F\, \frac{\beta^2}{2}\left(|H_V^0|^2+|H_A^0|^2\right)\,,\\ I_6^c = 2 F \frac{\beta\, m_\ell}{\sqrt{q^2}} {\rm Re} \left[ H_S^* H_V^0 \right]\,,\\ \]

where

\[ F=\frac{ \lambda^{1/2}\beta\, q^2}{3 \times 2^{5} \,\pi^3\, m_M^3}\,, \qquad \beta = \sqrt{1 - \frac{4 m_\ell^2}{q^2} }\,. \]

The final observables are hence build employing CP-averages $\Sigma_i$ or CP-asymmetries $\Delta_i$ of such angular coefficients; however, since on the experimental side the observables are averaged over $ q^2 $ bins, an integration of the coeffiecients over such bins has to be performed before they are combined in order to build the observables.

The class is organized as follows: after the parameters are updated in updateParameters() and the cache is checked in checkCache(), the form factor are build in the transverse basis in the functions f_plus(), f_0() and f_T() [Ball:2004ye]. They are consequentely translated in the helicity basis through the functions V_L(), V_R(), T_L(), T_R(), S_L() and S_R(). Form factors and parameters are combined together in the functions H_V(), H_A(), H_S() and H_P() in order to build the helicity amplitudes, which are consequentely combined to create the angular coefficients in the function I(). Those coefficients are used to create the CP averaged coefficients in the function Sigma() ad the CP asymmetric coefficients in the function Delta(). Form factors, CP averaged and asymmetric coefficients and hadronic contributions are integrated in the functions integrateSigma() and integrateDelta() in order to be further used to build the observables.

Definition at line 173 of file MPll.h.

Public Member Functions
gslpp::complex	DeltaC9_KD (double q2)

gslpp::complex	funct_g (double q2)

double	getSigma (int i, double q_2)
	The value of $ \Sigma_{i} $ from $q_{min}$ to $q_{max}$. More...

double	getwidth ()
	The width of the meson M. More...

gslpp::complex	H_A (double q2)
	The helicity amplitude $ H_A^{\lambda} $ . More...

gslpp::complex	h_lambda (double q2)
	The non-pertubative ccbar contributions to the helicity amplitudes. More...

gslpp::complex	H_nunu (double q2, QCD::lepton lep)
	The helicity amplitude $ H^{\lambda} $ for the invisible decay. More...

gslpp::complex	H_P (double q2)
	The helicity amplitude $ H_P^{\lambda} $ . More...

gslpp::complex	H_S (double q2)
	The helicity amplitude $ H_S^{\lambda} $ . More...

gslpp::complex	H_V (double q2)
	The helicity amplitude $ H_V^{\lambda} $ . More...

std::vector< std::string >	initializeMPllParameters ()
	A method for initializing the parameters necessary for MPll. More...

double	integrateDelta (int i, double q_min, double q_max)
	The integral of $ \Delta_{i} $ from $q_{min}$ to $q_{max}$. More...

double	integrateSigma (int i, double q_min, double q_max)
	The integral of $ \Sigma_{i} $ from $q_{min}$ to $q_{max}$. More...

double	integrateSigmaTree (double q_min, double q_max)
	The integral of $ \Sigma_{tree} $ from $q_{min}$ to $q_{max}$ (arxiv/2301.06990) More...

	MPll (const StandardModel &SM_i, QCD::meson meson_i, QCD::meson pseudoscalar_i, QCD::lepton lep_i)
	Constructor. More...

double	S_L (double q2)
	The helicity form factor $ S_L $. More...

gslpp::complex	T_L (double q2)
	The helicity form factor $ T_L^0 $. More...

gslpp::complex	V_L (double q2)
	The helicity form factor $ V_L^0 $. More...

virtual	~MPll ()
	Destructor. More...

Private Member Functions
gslpp::complex	Cpar (double q2)
	The correction $ C_{\parallel} $ from [Beneke:2001at] . More...

gslpp::complex	deltaC7_QCDF (double q2, bool spline=false)
	QCDF Correction from various BFS papers (hep-ph/0403185, hep-ph/0412400) and Greub et. al (arXiv:0810.4077).. More...

gslpp::complex	DeltaC9 (double q2)
	The total QCDF correction $ \Delta C_9 $ computed integrating over $ u $. More...

gslpp::complex	deltaC9_QCDF (double q2, bool spline=false)
	QCDF Correction from various BFS papers (hep-ph/0403185, hep-ph/0412400) and Greub et. al (arXiv:0810.4077).. More...

gslpp::complex	deltaTpar (double q2)
	The total correction $ \Delta \mathcal{T}^{\parallel} $ from [Beneke:2001at] . More...

gslpp::complex	F19 (double q2)
	The correction $ F_{19} $ from [Asatrian:2001de]. More...

gslpp::complex	F27 (double q2)
	The correction $ F_{27} $ from [Asatrian:2001de]. More...

gslpp::complex	F29 (double q2)
	The correction $ F_{29} $ from [Asatrian:2001de]. More...

gslpp::complex	F87 (double q2)
	The correction $ F_{87} $ from [Asatrian:2001de]. More...

double	F89 (double q2)
	The correction $ F_{89} $ from [Asatrian:2001de]. More...

gslpp::complex	fDeltaC9 (double q2)
	The total QCDF correction $ \Delta C_9 $ computed fitting over $ u $. More...

void	fit_DeltaC9_mumu ()
	The fitting routine for the QCDF correction $ \Delta C_9^0 $ in the muon channel. More...

double	getDelta1c (double q2)
	The CP asymmetry $ \Delta_{1s} $ . More...

double	getDelta2c (double q2)
	The CP asymmetry $ \Delta_{2s} $ . More...

double	getintegratedSigmaTree ()
	The integral of $ \Sigma_{tree} $ from 0 to $q_{cut}$. More...

double	getSigma1c (double q2)
	The CP average $ \Sigma_{1s} $ . More...

double	getSigma2c (double q2)
	The CP average $ \Sigma_{2s} $ . More...

double	getSigma6c (double q2)
	The CP average $ \Sigma_{6s} $ . More...

gslpp::complex	I1 (double u, double q2)
	The $ I_1 $ function from [Beneke:2001at] . More...

double	imDC9fit (double x, double p)
	The fit function for the imaginary part of the QCDF correction $ \Delta C_9^{\lambda} $. More...

double	Integrand_ImTpar_pm (double up)
	The sum of Integrand_ImTparplus() and Integrand_ImTparminus(). More...

double	Integrand_ImTparminus (double up)
	The imaginary part of the integral involving $ T^{\parallel}_- $ at fixed $ q^2 $, according to [Beneke:2001at] . More...

double	Integrand_ImTparplus (double up)
	The imaginary part of the integral involving $ T^{\parallel}_+ $ at fixed $ q^2 $, according to [Beneke:2001at] . More...

double	Integrand_ReTpar_pm (double up)
	The sum of Integrand_ReTparplus() and Integrand_ReTparminus(). More...

double	Integrand_ReTparminus (double up)
	The real part of the integral involving $ T^{\parallel}_- $ at fixed $ q^2 $, according to [Beneke:2001at] . More...

double	Integrand_ReTparplus (double up)
	The real part of the integral involving $ T^{\parallel}_+ $ at fixed $ q^2 $, according to [Beneke:2001at] . More...

double	reDC9fit (double x, double p)
	The fit function for the real part of the QCDF correction $ \Delta C_9^{\lambda} $. More...

double	SigmaTree (double q2)

void	spline_QCDF_func ()

gslpp::complex	Tparminus (double u, double q2)
	The $ T^{\parallel}_- $ function from [Khodjamirian:2012rm] . More...

gslpp::complex	Tparplus (double u, double q2)
	The $ T^{\parallel}_+ $ function from [Khodjamirian:2012rm] . More...

Private Attributes
double	ale

double	alpha_s_mub

bool	dispersion = false

bool	FixedWCbtos = false

double	GF

QCD::lepton	lep

double	Mb

double	mb_pole

double	Mc

double	mc_pole

QCD::meson	meson

double	mJ2

double	Mlep

double	MM

double	MP

bool	MPll_DM_flag

bool	MPll_GRvDV_flag

bool	MPll_Lattice_flag

std::vector< std::string >	mpllParameters

double	Ms

double	mu_b

double	mu_h

std::unique_ptr< F_1 >	myF_1

std::unique_ptr< F_2 >	myF_2

const StandardModel &	mySM

bool	NeutrinoTree_flag

QCD::meson	pseudoscalar

double	spectator_charge

double	width

Constructor & Destructor Documentation

◆ MPll()

MPll::MPll	(	const StandardModel &	SM_i,
		QCD::meson	meson_i,
		QCD::meson	pseudoscalar_i,
		QCD::lepton	lep_i
	)

Constructor.

Parameters

[in]	SM_i	a reference to an object of type StandardModel
[in]	meson_i	initial meson of the decay
[in]	pseudoscalar_i	final pseudoscalar meson of the decay
[in]	lep_i	final leptons of the decay

Definition at line 20 of file MPll.cpp.

: mySM(SM_i), myF_1(new F_1()), myF_2(new F_2()),
fplus_lat_cache(3, 0.),
fT_lat_cache(3, 0.),
f0_lat_cache(3, 0.),
fplus_cache(2, 0.),
fT_cache(2, 0.),
k2_cache(2, 0.),
SL_cache(2, 0.),
N_cache(3, 0.),
Ycache(2, 0.),
H_V0cache(2, 0.),
H_Scache(2, 0.),
H_P_cache(4, 0.),
Itree_cache(3, 0.),
T_cache(5, 0.)
{
    lep = lep_i;
    meson = meson_i;
    pseudoscalar = pseudoscalar_i;
    dispersion = true;
    FixedWCbtos = false;
    MPll_Lattice_flag = false;
    MPll_GRvDV_flag = false;
    MPll_DM_flag = false;
    NeutrinoTree_flag = false;
    mJ2 = 3.096 * 3.096;
 
    I0_updated = 0;
    I2_updated = 0;
    I8_updated = 0;
    Itree_updated = 0;
 
    VL_updated = 0;
    TL_updated = 0;
    SL_updated = 0;
 
    deltaTparpupdated = 0;
    deltaTparmupdated = 0;
 
    w_sigma = gsl_integration_cquad_workspace_alloc(100);
    w_delta = gsl_integration_cquad_workspace_alloc(100);
    w_sigmaTree = gsl_integration_cquad_workspace_alloc(100);
    w_DTPPR = gsl_integration_cquad_workspace_alloc(100);
 
    acc_Re_deltaC7_QCDF = gsl_interp_accel_alloc();
    acc_Im_deltaC7_QCDF = gsl_interp_accel_alloc();
    acc_Re_deltaC9_QCDF = gsl_interp_accel_alloc();
    acc_Im_deltaC9_QCDF = gsl_interp_accel_alloc();
 
    spline_Re_deltaC7_QCDF = gsl_spline_alloc(gsl_interp_cspline, GSL_INTERP_DIM_DC);
    spline_Im_deltaC7_QCDF = gsl_spline_alloc(gsl_interp_cspline, GSL_INTERP_DIM_DC);
    spline_Re_deltaC9_QCDF = gsl_spline_alloc(gsl_interp_cspline, GSL_INTERP_DIM_DC);
    spline_Im_deltaC9_QCDF = gsl_spline_alloc(gsl_interp_cspline, GSL_INTERP_DIM_DC);
}

◆ ~MPll()

MPll::~MPll ( )

virtual

Destructor.

Definition at line 76 of file MPll.cpp.

77{

78}

Member Function Documentation

◆ Cpar()

gslpp::complex MPll::Cpar ( double q2 )

private

The correction $ C_{\parallel} $ from [Beneke:2001at] .

Parameters

[in] q2 $q^2$ of the decay

Returns: $ C_{\parallel} $

Definition at line 1109 of file MPll.cpp.

{
    return -(C_2L_bar * F27(q2) + C_8L * F87(q2) + MM / 2. / Mb *
            (C_2L_bar * F29(q2) + 2. * C_1L_bar * (F19(q2) + F29(q2) / 6.) + C_8L * F89(q2)));
}

◆ deltaC7_QCDF()

gslpp::complex MPll::deltaC7_QCDF	(	double	q2,
		bool	spline = `false`
	)

private

QCDF Correction from various BFS papers (hep-ph/0403185, hep-ph/0412400) and Greub et. al (arXiv:0810.4077)..

Parameters

q2	$q^2$ of the decay

Returns: $ \Delta C_{7}^{QCDF} $

Definition at line 1198 of file MPll.cpp.

{
#if SPLINE
    if (spline) return gsl_spline_eval(spline_Re_deltaC7_QCDF, q2, acc_Re_deltaC7_QCDF);
#endif
 
    double muh = mu_b / mb_pole;
    double z = mc_pole * mc_pole / mb_pole / mb_pole;
    double sh = q2 / mb_pole / mb_pole;
    double sh2 = sh*sh;
 
    //gslpp::complex A_Sdl = A_Seidel(q2, mb_pole*mb_pole); /* hep-ph/0403185v2.*/
    //gslpp::complex Fu_17 = -A_Sdl; /* sign different from hep-ph/0403185v2 but consistent with hep-ph/0412400 */
    //gslpp::complex Fu_27 = 6. * A_Sdl; /* sign different from hep-ph/0403185v2 but consistent with hep-ph/0412400 */
    gslpp::complex F_17 = myF_1->F_17re(muh, z, sh, 20) + gslpp::complex::i() * myF_1->F_17im(muh, z, sh, 20); /*q^2 = 0 gives nan. Independent of how small q^2 is. arXiv:0810.4077*/
    gslpp::complex F_27 = myF_2->F_27re(muh, z, sh, 20) + gslpp::complex::i() * myF_2->F_27im(muh, z, sh, 20); /*q^2 = 0 gives nan. Independent of how small q^2 is. arXiv:0810.4077*/
    gslpp::complex F_87 = F87_0 + F87_1 * sh + F87_2 * sh2 + F87_3 * sh * sh2 - 8. / 9. * log(sh) * (sh + sh2 + sh * sh2);
 
    gslpp::complex delta = C_1 * F_17 + C_2 * F_27;
    gslpp::complex delta_t = C_8 * F_87 + delta;
    //gslpp::complex delta_u = delta + C_1 * Fu_17 + C_2 * Fu_27;
 
    return -alpha_s_mub / (4. * M_PI) * (delta_t /*- lambda_u / lambda_t * delta_u */);
}

◆ DeltaC9()

gslpp::complex MPll::DeltaC9 ( double q2 )

private

The total QCDF correction $ \Delta C_9 $ computed integrating over $ u $.

Parameters

[in] q2 $q^2$ of the decay

Returns: $ \Delta C_9 $

Definition at line 1193 of file MPll.cpp.

{
    return deltaTpar(q2);
}

◆ DeltaC9_KD()

gslpp::complex MPll::DeltaC9_KD ( double q2 )

Definition at line 1315 of file MPll.cpp.

{
    return ((Delta_C9 + r_1 * q2 / mJ2) / (1. - r_2 * q2 / mJ2) - (3. * (-0.267) + 1.117) * funct_g(q2))*exp_Phase;
    /* C_1 = -0.267 and C_2 = 1.117 in KMPW */
}

◆ deltaC9_QCDF()

gslpp::complex MPll::deltaC9_QCDF	(	double	q2,
		bool	spline = `false`
	)

private

QCDF Correction from various BFS papers (hep-ph/0403185, hep-ph/0412400) and Greub et. al (arXiv:0810.4077)..

Parameters

q2	$q^2$ of the decay

Returns: $ \Delta C_{9}^{QCDF} $

Definition at line 1223 of file MPll.cpp.

{
#if SPLINE
    if (spline) return gsl_spline_eval(spline_Re_deltaC9_QCDF, q2, acc_Re_deltaC9_QCDF);
#endif
    double muh = mu_b / mb_pole;
    double z = mc_pole * mc_pole / mb_pole / mb_pole;
    double sh = q2 / mb_pole / mb_pole;
    double sh2 = sh*sh;
 
    //gslpp::complex B_Sdl = B_Seidel(q2, mb_pole*mb_pole); /* hep-ph/0403185v2.*/
    //gslpp::complex C_Sdl = C_Seidel(q2); /* hep-ph/0403185v2.*/
    //gslpp::complex Fu_19 = -(B_Sdl + 4. * C_Sdl); /* sign different from hep-ph/0403185v2 but consistent with hep-ph/0412400 */
    //gslpp::complex Fu_29 = -(-6. * B_Sdl + 3. * C_Sdl); /* sign different from hep-ph/0403185v2 but consistent with hep-ph/0412400 */
    gslpp::complex F_19 = myF_1->F_19re(muh, z, sh, 20) + gslpp::complex::i() * myF_1->F_19im(muh, z, sh, 20); /*q^2 = 0 gives nan. Independent of how small q^2 is. arXiv:0810.4077*/
    gslpp::complex F_29 = myF_2->F_29re(muh, z, sh, 20) + gslpp::complex::i() * myF_2->F_29im(muh, z, sh, 20); /*q^2 = 0 gives nan. Independent of how small q^2 is. arXiv:0810.4077*/
    gslpp::complex F_89 = (F89_0 + F89_1 * sh + F89_2 * sh2 + F89_3 * sh * sh2 + 16. / 9. * log(sh) * (1. + sh + sh2 + sh * sh2));
 
    gslpp::complex delta = C_1 * F_19 + C_2 * F_29;
    gslpp::complex delta_t = C_8 * F_89 + delta;
    //gslpp::complex delta_u = delta + C_1 * Fu_19 + C_2 * Fu_29;
 
    return -alpha_s_mub / (4. * M_PI) * (delta_t /*- lambda_u / lambda_t * delta_u*/);
}

◆ deltaTpar()

gslpp::complex MPll::deltaTpar ( double q2 )

private

The total correction $ \Delta \mathcal{T}^{\parallel} $ from [Beneke:2001at] .

Parameters

[in] q2 $q^2$ of the decay

Returns: $ \Delta \mathcal{T}^{\parallel} $

Definition at line 1115 of file MPll.cpp.

{
    tmpq2 = q2;
 
    //old_handler = gsl_set_error_handler_off();
 
    if (deltaTparpCached[q2] == 0) {
 
        DTPPR = convertToGslFunction(bind(&MPll::Integrand_ReTpar_pm, &(*this), _1));
        if (gsl_integration_cquad(&DTPPR, 0., 1., 1.e-2, 1.e-1, w_DTPPR, &avaDTPPR, &errDTPPR, NULL) != 0) return std::numeric_limits<double>::quiet_NaN();
        double ReTppint = avaDTPPR;
 
        DTPPR = convertToGslFunction(bind(&MPll::Integrand_ImTpar_pm, &(*this), _1));
        if (gsl_integration_cquad(&DTPPR, 0., 1., 1.e-2, 1.e-1, w_DTPPR, &avaDTPPR, &errDTPPR, NULL) != 0) return std::numeric_limits<double>::quiet_NaN();
        double ImTppint = avaDTPPR;
 
        cacheDeltaTparp[q2] = (ReTppint + gslpp::complex::i() * ImTppint);
        deltaTparpCached[q2] = 1;
    }
 
    //gsl_set_error_handler(old_handler);
 
    return deltaT_0 * Cpar(q2) + deltaT_1par * cacheDeltaTparp[q2] / f_plus(q2);
}

◆ F19()

gslpp::complex MPll::F19 ( double q2 )

private

The correction $ F_{19} $ from [Asatrian:2001de].

Parameters

[in] q2 $q^2$ of the decay

Returns: $ F_{19} $

Definition at line 1063 of file MPll.cpp.

{
    double s = q2 / Mb2;
    double s2 = s*s;
    double Ls = log(s);
    double Lc = log(Mc / Mb);
    double Lm = log(mu_b / Mb);
    gslpp::complex i = gslpp::complex::i();
    return (-1424. / 729. + 16. / 243. * i * M_PI + 64. / 27. * Lc)*Lm - 16. / 243. * Lm * Ls + (16. / 1215. - 32. / 135. / Mc2 * Mb2) * Lm * s
            + (4. / 2835. - 8. / 315. / Mc2 * Mb2 / Mc2 * Mb2) * Lm * s2 + (16. / 76545. - 32. / 8505 / Mc2 * Mb2 / Mc2 * Mb2 / Mc2 * Mb2) * Lm * s * s2 - 256. / 243. * Lm * Lm
            + (-11.65 + 0.18223 * i + (-24.709 - 0.13474 * i) * s + (-43.588 - 0.4738 * i) * s2 + (-86.22 - 1.3542 * i) * s * s2
            + (-0.080959 - 0.051864 * i + (-0.036585 + 0.024753 * i) * s + (-0.021692 + 0.036925 * i) * s2 + (0.013282 + 0.052023 * i) * s * s2) * Ls);
}

◆ F27()

gslpp::complex MPll::F27 ( double q2 )

private

The correction $ F_{27} $ from [Asatrian:2001de].

Parameters

[in] q2 $q^2$ of the decay

Returns: $ F_{27} $

Definition at line 1077 of file MPll.cpp.

{
    double s = q2 / Mb2;
    double s2 = s*s;
    double Ls = log(s);
    gslpp::complex i = gslpp::complex::i();
    return F27_0 + F27_1 * s + F27_2 * s2 + F27_3 * s * s2 + F27_L1_1 * Ls * s + F27_L1_2 * Ls * s2 + F27_L1_3 * Ls * s * s2;
}

◆ F29()

gslpp::complex MPll::F29 ( double q2 )

private

The correction $ F_{29} $ from [Asatrian:2001de].

Parameters

[in] q2 $q^2$ of the decay

Returns: $ F_{29} $

Definition at line 1086 of file MPll.cpp.

{
    double s = q2 / Mb2;
    double s2 = s*s;
    double Ls = log(s);
    gslpp::complex i = gslpp::complex::i();
    return F29_0 + F29_L1 * Ls + F29_1 * s + F29_2 * s2 + F29_3 * s * s2 + F29_L1_1 * Ls * s + F29_L1_2 * Ls * s2 + F29_L1_3 * Ls * s2 *s;
}

◆ F87()

gslpp::complex MPll::F87 ( double q2 )

private

The correction $ F_{87} $ from [Asatrian:2001de].

Parameters

[in] q2 $q^2$ of the decay

Returns: $ F_{87} $

Definition at line 1095 of file MPll.cpp.

{
    double s = q2 / Mb2;
    double s2 = s*s;
    return F87_0 + F87_1 * s + F87_2 * s2 + F87_3 * s * s2 - 0.888889 * log(s)*(1. + s + s2 + s * s2);
}

◆ F89()

double MPll::F89 ( double q2 )

private

The correction $ F_{89} $ from [Asatrian:2001de].

Parameters

[in] q2 $q^2$ of the decay

Returns: $ F_{89} $

Definition at line 1102 of file MPll.cpp.

{
    double s = q2 / Mb2;
    double s2 = s*s;
    return F89_0 + F89_1 * s + F89_2 * s2 + F89_3 * s * s2 + 1.77778 * log(s)*(1. + s + s2 + s * s2);
}

◆ fDeltaC9()

gslpp::complex MPll::fDeltaC9 ( double q2 )

private

The total QCDF correction $ \Delta C_9 $ computed fitting over $ u $.

Parameters

[in] q2 $q^2$ of the decay

Returns: $ \Delta C_9 $

Definition at line 1184 of file MPll.cpp.

{
    if (q2 < MPllSWITCH) return (reDC9fit(&q2, const_cast<double *> (refres->GetParams()))
            + gslpp::complex::i() * imDC9fit(&q2, const_cast<double *> (imfres->GetParams())));
    else return (reDC9fit(&q2, const_cast<double *> (refres->GetParams()))
            + gslpp::complex::i() * imDC9fit(&q2, const_cast<double *> (imfres->GetParams()))) / q2;
 
}

◆ fit_DeltaC9_mumu()

void MPll::fit_DeltaC9_mumu ( )

private

The fitting routine for the QCDF correction $ \Delta C_9^0 $ in the muon channel.

Definition at line 1152 of file MPll.cpp.

{
    int dim = 0;
    for (double i = 0.1; i < MPllSWITCH; i += 0.4) {
        double q2tmp = i;
        myq2.push_back(q2tmp);
        ReDeltaC9.push_back((deltaTpar(q2tmp)).real());
        ImDeltaC9.push_back((deltaTpar(q2tmp)).imag());
        dim++;
    }
    for (double i = MPllSWITCH; i < 8.2; i += 0.4) {
        double q2tmp = i;
        myq2.push_back(q2tmp);
        ReDeltaC9.push_back(q2tmp * (deltaTpar(q2tmp)).real());
        ImDeltaC9.push_back(q2tmp * (deltaTpar(q2tmp)).imag());
        dim++;
    }
 
    gr1 = TGraph(dim, myq2.data(), ReDeltaC9.data());
    gr2 = TGraph(dim, myq2.data(), ImDeltaC9.data());
 
    reffit = TF1("reffit", this, &MPll::reDC9fit, 0, 8.1, 7);
    imffit = TF1("imffit", this, &MPll::imDC9fit, 0, 8.1, 8);
 
    refres = gr1.Fit(&reffit, "SQN0+rob=0.99");
    imfres = gr2.Fit(&imffit, "SQN0+rob=0.99");
 
    ReDeltaC9.clear();
    ImDeltaC9.clear();
    myq2.clear();
}

◆ funct_g()

gslpp::complex MPll::funct_g ( double q2 )

Definition at line 1307 of file MPll.cpp.

{
    if (q2 < 4. * Mc * Mc)
        return -8. / 9. * log(Mc / Mb) + 8. / 27. + 16. / 9. * Mc * Mc / q2 - 4. / 9. * (2. + 4. * Mc * Mc / q2) * (sqrt(4. * Mc * Mc / q2 - 1.) * atan(1. / sqrt(4. * Mc * Mc / q2 - 1.)));
    else
        return -8. / 9. * log(Mc / Mb) + 8. / 27. + 16. / 9. * Mc * Mc / q2 - 4. / 9. * (2. + 4. * Mc * Mc / q2) * (sqrt(1. - 4. * Mc * Mc / q2) * (log(1. + sqrt(1. - 4. * Mc * Mc / q2) / sqrt(4. * Mc * Mc / q2)) - gslpp::complex::i() * M_PI_2));
}

◆ getDelta1c()

double MPll::getDelta1c ( double q2 )

inlineprivate

The CP asymmetry $ \Delta_{1s} $ .

Parameters

[in] q2 $q^2$ of the decay

Returns: $ \Delta_{1c} $

Definition at line 1046 of file MPll.h.

    {
        return Delta(0, q2);
    };

◆ getDelta2c()

double MPll::getDelta2c ( double q2 )

inlineprivate

The CP asymmetry $ \Delta_{2s} $ .

Parameters

[in] q2 $q^2$ of the decay

Returns: $ \Delta_{2s} $

Definition at line 1056 of file MPll.h.

    {
        return Delta(2, q2);
    };

◆ getintegratedSigmaTree()

double MPll::getintegratedSigmaTree ( )

private

The integral of $ \Sigma_{tree} $ from 0 to $q_{cut}$.

Returns: $ <\Sigma{tree}> $

Definition at line 1564 of file MPll.cpp.

{
    return mySM.getMesons(meson).getLifetime() / HCUT * GF4 * VusVub_abs2 * fP2 * fM2 / (128. * M_PI2 * MM3 * Gammatau) * mtau * (mtau2 - MP2) * (mtau2 - MP2) * (MM2 - mtau2) * (MM2 - mtau2);
}

◆ getSigma()

double MPll::getSigma	(	int	i,
		double	q_2
	)

The value of $ \Sigma_{i} $ from $q_{min}$ to $q_{max}$.

Parameters

[in]	i	index of the angular coefficient $ I_{i} $
[in]

_form#752 value of the function

Returns: $ <\Sigma_{i}> $

Definition at line 1471 of file MPll.cpp.

{
    updateParameters();
 
    switch (i) {
        case 0:
            return getSigma1c(q_2);
            break;
        case 2:
            return getSigma2c(q_2);
            break;
        case 8:
            return getSigma6c(q_2);
            break;
        default:
            std::stringstream out;
            out << i;
            throw std::runtime_error("MPll::getSigma: index " + out.str() + " not implemented");
    }
}

◆ getSigma1c()

double MPll::getSigma1c ( double q2 )

inlineprivate

The CP average $ \Sigma_{1s} $ .

Parameters

[in] q2 $q^2$ of the decay

Returns: $ \Sigma_{1c} $

Definition at line 1016 of file MPll.h.

    {
        return I_1c(q2);
    };

◆ getSigma2c()

double MPll::getSigma2c ( double q2 )

inlineprivate

The CP average $ \Sigma_{2s} $ .

Parameters

[in] q2 $q^2$ of the decay

Returns: $ \Sigma_{2c} $

Definition at line 1026 of file MPll.h.

    {
        return I_2c(q2);
    };

◆ getSigma6c()

double MPll::getSigma6c ( double q2 )

inlineprivate

The CP average $ \Sigma_{6s} $ .

Parameters

[in] q2 $q^2$ of the decay

Returns: $ \Sigma_{6c} $

Definition at line 1036 of file MPll.h.

    {
        return I_6c(q2);
    };

◆ getwidth()

double MPll::getwidth ( )

inline

The width of the meson M.

Returns: $ \Gamma_M $

Definition at line 302 of file MPll.h.

                     {
        updateParameters();
        return width;
    }

◆ H_A()

gslpp::complex MPll::H_A ( double q2 )

The helicity amplitude $ H_A^{\lambda} $ .

Parameters

[in] q2 $q^2$ of the decay

Returns: $ H_A^{\lambda} $

Definition at line 1345 of file MPll.cpp.

{
    return (-C_10 + etaP * pow(-1, angmomP) * C_10p) *V_L(q2);
}

◆ h_lambda()

gslpp::complex MPll::h_lambda ( double q2 )

The non-pertubative ccbar contributions to the helicity amplitudes.

Parameters

hel	helicity
q2	$q^2$

Returns: $h_{hel}(q^2)$

Definition at line 1321 of file MPll.cpp.

{
    if (!dispersion) {
//        if (q2 <= 1.) return 1.3e-4/MM2 * (1. + gslpp::complex::i()) / sqrt(2.);
//        else {
//            h_2 = (-sixteenM_PI2*4.9e-7/MM2 * (1. + gslpp::complex::i()) / sqrt(2.) - h_1 / MM2 * V_L(1.) - h_2 * sixteenM_PI2) / twoMboMM / T_L(1.);
//            h_2 = 1.3e-4/MM2 * (1. + gslpp::complex::i()) / sqrt(2.) - h_1 * V_L(1.)/sixteenM_PI2/MM2;
            //else return 4.9e-7/MM2 + h_1 * (q2 * V_L(q2) - T_L(q2). * V_L(1)/T_L(1.)) / MM2  / sixteenM_PI2;
            
            // return (twoMboMM * h_0 * T_L(q2) + h_1 * q2 / MM2 * V_L(q2)) / sixteenM_PI2 + h_2 * q2 * sqrt(q2);
            return (h_1 + h_2 * q2 ) * sqrt(q2) * sqrt(lambda(q2)) / MM3 / 2.;
//        }
    } else {
        return -q2 / (MM2 * sixteenM_PI2) * V_L(q2) * DeltaC9_KD(q2);
    }
}

◆ H_nunu()

gslpp::complex MPll::H_nunu	(	double	q2,
		QCD::lepton	lep
	)

The helicity amplitude $ H^{\lambda} $ for the invisible decay.

Parameters

[in]	q2	$q^2$ of the decay
[in]	lep	final leptons of the decay

Returns: $ H^{\lambda} $

Definition at line 1360 of file MPll.cpp.

{
    if (lep == QCD::NEUTRINO_1) 
        return -(C_L_nunu_e - etaP * pow(-1, angmomP) * C_R_nunu_e) * V_L(q2);
    else if (lep == QCD::NEUTRINO_2) 
        return -(C_L_nunu_mu - etaP * pow(-1, angmomP) * C_R_nunu_mu) * V_L(q2);
    else if (lep == QCD::NEUTRINO_3) 
        return -(C_L_nunu_tau - etaP * pow(-1, angmomP) * C_R_nunu_tau) * V_L(q2);
    else throw std::runtime_error("MPll::H_nunu: lepton not supported");
}

◆ H_P()

gslpp::complex MPll::H_P ( double q2 )

The helicity amplitude $ H_P^{\lambda} $ .

Parameters

[in] q2 $q^2$ of the decay

Returns: $ H_P^{\lambda} $

Definition at line 1355 of file MPll.cpp.

{
    return ( MboMW * (C_P - etaP * pow(-1, angmomP) * C_Pp) + twoMlepMb / q2 * (C_10 * (1. + etaP * pow(-1, angmomP) * MsoMb) - C_10p * (etaP * pow(-1, angmomP) + MsoMb))) * S_L(q2);
}

◆ H_S()

gslpp::complex MPll::H_S ( double q2 )

The helicity amplitude $ H_S^{\lambda} $ .

Parameters

[in] q2 $q^2$ of the decay

Returns: $ H_S^{\lambda} $

Definition at line 1350 of file MPll.cpp.

{
    return MboMW * (C_S - etaP * pow(-1, angmomP) * C_Sp) * S_L(q2);
}

◆ H_V()

gslpp::complex MPll::H_V ( double q2 )

The helicity amplitude $ H_V^{\lambda} $ .

Parameters

[in] q2 $q^2$ of the decay

Returns: $ H_V^{\lambda} $

Definition at line 1338 of file MPll.cpp.

{
    return -((C_9 + deltaC9_QCDF(q2, SPLINE) + Y(q2) /*+ fDeltaC9(q2)*/ - etaP * pow(-1, angmomP) * C_9p) * V_L(q2)
            + MM2 / q2 * (twoMboMM * (C_7 + deltaC7_QCDF(q2, SPLINE) - etaP * pow(-1, angmomP) * C_7p) * T_L(q2)
            - sixteenM_PI2 * h_lambda(q2)));
}

◆ I1()

gslpp::complex MPll::I1	(	double	u,
		double	q2
	)

private

The $ I_1 $ function from [Beneke:2001at] .

Parameters

[in]	u	dummy variable to be integrated out
[in]	q2	$q^2$ of the decay

Returns: $ I_1 $

Definition at line 973 of file MPll.cpp.

{
    std::pair<double, double > uq2 = std::make_pair(u, q2);
 
    if (I1Cached[uq2] == 0) {
        ubar = 1. - u;
        xp = .5 + sqrt(0.25 - (Mc2 - gslpp::complex::i()*1.e-10) / (ubar * MM2 + u * q2));
        xm = .5 - sqrt(0.25 - (Mc2 - gslpp::complex::i()*1.e-10) / (ubar * MM2 + u * q2));
        yp = .5 + sqrt(0.25 - (Mc2 - gslpp::complex::i()*1.e-10) / q2);
        ym = .5 - sqrt(0.25 - (Mc2 - gslpp::complex::i()*1.e-10) / q2);
        L1xp = log(1. - 1. / xp) * log(1. - xp) - M_PI2osix + dilog(xp / (xp - 1.));
        L1xm = log(1. - 1. / xm) * log(1. - xm) - M_PI2osix + dilog(xm / (xm - 1.));
        L1yp = log(1. - 1. / yp) * log(1. - yp) - M_PI2osix + dilog(yp / (yp - 1.));
        L1ym = log(1. - 1. / ym) * log(1. - ym) - M_PI2osix + dilog(ym / (ym - 1.));
 
        cacheI1[uq2] = 1. + twoMc2 / ubar / (MM2 - q2)*(L1xp + L1xm - L1yp - L1ym);
        I1Cached[uq2] = 1;
    }
 
    return cacheI1[uq2];
}

◆ imDC9fit()

double MPll::imDC9fit	(	double *	x,
		double *	p
	)

private

The fit function for the imaginary part of the QCDF correction $ \Delta C_9^{\lambda} $.

Parameters

[in]	x	fit variable
[in]	p	fit parameters vector

Returns: $ f_{Im \Delta C_9^{\lambda}} $

Definition at line 1145 of file MPll.cpp.

{
    return p[0] / x[0] + p[1] + p[2] * x[0] + p[3] * x[0] * x[0] + p[4] * x[0] * x[0] * x[0] + p[5] * x[0] * x[0] * x[0] * x[0] + p[6] * x[0] * x[0] * x[0] * x[0] * x[0] + p[7] * x[0] * x[0] * x[0] * x[0] * x[0] * x[0];
 
    //double thr = 4.*Mc2;
}

◆ initializeMPllParameters()

std::vector< std::string > MPll::initializeMPllParameters ( )

A method for initializing the parameters necessary for MPll.

Returns: the vector of MPll specific parameters

Definition at line 80 of file MPll.cpp.

{
    dispersion = mySM.getFlavour().getFlagUseDispersionRelation();
    FixedWCbtos = mySM.getFlavour().getFlagFixedWCbtos();
    MPll_Lattice_flag = mySM.getFlavour().getFlagMPll_FNALMILC();
    MPll_GRvDV_flag = mySM.getFlavour().getFlagMPll_GRvDV();
    MPll_DM_flag = mySM.getFlavour().getFlagMPll_DM();
    NeutrinoTree_flag = mySM.getFlavour().getFlagNeutrinoTree();
 
    if (pseudoscalar == StandardModel::K_P || pseudoscalar == StandardModel::K_0) {
        if (MPll_Lattice_flag) mpllParameters = make_vector<std::string>()
            << "b_0_fplus" << "b_1_fplus" << "b_2_fplus" << "m_fit_fplus_lat"
            << "b_0_fT" << "b_1_fT" << "b_2_fT" << "m_fit_fT_lat"
            << "b_0_f0" << "b_1_f0" << "b_2_f0" << "m_fit_f0_lat" ;
        else if (MPll_GRvDV_flag) mpllParameters = make_vector<std::string>()
            << "b_0_fplus" << "b_1_fplus" << "b_2_fplus" << "m_fit_fplus_lat"
            << "b_0_fT" << "b_1_fT" << "b_2_fT" << "m_fit_fT_lat"
            << "b_1_f0" << "b_2_f0" << "m_fit_f0_lat" ;
        else if (MPll_DM_flag) mpllParameters = make_vector<std::string>()
            << "b_0_fplus" << "b_1_fplus" << "b_2_fplus" << "m_fit_fplus_lat"
            << "b_0_fT" << "b_1_fT" << "b_2_fT" << "m_fit_fT_lat"
            << "b_1_f0" << "b_2_f0" ;
        else mpllParameters = make_vector<std::string>()
            << "r_1_fplus" << "r_2_fplus" << "m_fit2_fplus" << "r_1_fT" << "r_2_fT" << "m_fit2_fT" << "r_2_f0" << "m_fit2_f0";
    } else {
        std::stringstream out;
        out << pseudoscalar;
        throw std::runtime_error("MPll: pseudoscalar " + out.str() + " not implemented");
    }
 
    if (lep != QCD::NEUTRINO_1){
        if (dispersion) {
            mpllParameters.insert(mpllParameters.end(), { "r1_BK", "r2_BK", "deltaC9_BK", "phDC9_BK" });
        } else {
#if NFPOLARBASIS_MPLL
            mpllParameters.insert(mpllParameters.end(), { "absh_0_MP", "argh_0_MP", "absh_1_MP", "argh_1_MP", "absh_2_MP", "argh_2_MP", "Delta_C7_U", "Delta_C9_U" });
#else
            mpllParameters.insert(mpllParameters.end(), { "reh_0_MP", "imh_0_MP", "reh_1_MP", "imh_1_MP", "reh_2_MP", "imh_2_MP", "Delta_C7_U", "Delta_C9_U" });
#endif
        }
    }
 
    if (FixedWCbtos) 
        if (lep != QCD::NEUTRINO_1) mpllParameters.insert(mpllParameters.end(), { "C7_SM", "C9_SM", "C10_SM" });
        else mpllParameters.insert(mpllParameters.end(), { "CLnunu_SM" });
    mySM.initializeMeson(meson);
    mySM.initializeMeson(pseudoscalar);
    return mpllParameters;
}

◆ Integrand_ImTpar_pm()

double MPll::Integrand_ImTpar_pm ( double up )

private

The sum of Integrand_ImTparplus() and Integrand_ImTparminus().

Parameters

[in] up dummy variable to be integrated out

Returns: $ Im T^{\parallel}_+ \Phi^{\parallel} + Im T^{\parallel}_- \Phi^{\parallel}$

Definition at line 1053 of file MPll.cpp.

{
    return Integrand_ImTparplus(up) + Integrand_ImTparminus(up);
}

◆ Integrand_ImTparminus()

double MPll::Integrand_ImTparminus ( double up )

private

The imaginary part of the integral involving $ T^{\parallel}_- $ at fixed $ q^2 $, according to [Beneke:2001at] .

Parameters

[in] up dummy variable to be integrated out

Returns: $ Im T^{\parallel}_- \Phi^{\parallel}$

Definition at line 1042 of file MPll.cpp.

{
    double Lambdaplus = mySM.getMesons(meson).getLambdaM();
    gslpp::complex Lambdamin = exp(-tmpq2 / MM / Lambdaplus) / Lambdaplus * (-gsl_sf_expint_Ei(tmpq2 / MM / Lambdaplus) + gslpp::complex::i() * M_PI);
 
    double u = up;
    return ((Tparminus(u, tmpq2)*6. * u * (1. - u)*
            (1 + threeGegen0 * (2. * u - 1)
            + threeGegen1otwo * ((10. * u - 5.)*(2. * u - 1.) - 1.))) / Lambdamin).imag();
}

◆ Integrand_ImTparplus()

double MPll::Integrand_ImTparplus ( double up )

private

The imaginary part of the integral involving $ T^{\parallel}_+ $ at fixed $ q^2 $, according to [Beneke:2001at] .

Parameters

[in] up dummy variable to be integrated out

Returns: $ Im T^{\parallel}_+ \Phi^{\parallel}$

Definition at line 1023 of file MPll.cpp.

{
    double u = up;
    return ((Tparplus(u, tmpq2)*6. * u * (1. - u)*
            (1 + threeGegen0 * (2. * u - 1)
            + threeGegen1otwo * ((10. * u - 5.)*(2. * u - 1.) - 1.))) / mySM.getMesons(meson).getLambdaM()).imag();
}

◆ Integrand_ReTpar_pm()

double MPll::Integrand_ReTpar_pm ( double up )

private

The sum of Integrand_ReTparplus() and Integrand_ReTparminus().

Parameters

[in] up dummy variable to be integrated out

Returns: $ Re T^{\parallel}_+ \Phi^{\parallel} + Re T^{\parallel}_- \Phi^{\parallel}$

Definition at line 1058 of file MPll.cpp.

{
    return Integrand_ReTparplus(up) + Integrand_ReTparminus(up);
}

◆ Integrand_ReTparminus()

double MPll::Integrand_ReTparminus ( double up )

private

The real part of the integral involving $ T^{\parallel}_- $ at fixed $ q^2 $, according to [Beneke:2001at] .

Parameters

[in] up dummy variable to be integrated out

Returns: $ Re T^{\parallel}_- \Phi^{\parallel}$

Definition at line 1031 of file MPll.cpp.

{
    double Lambdaplus = mySM.getMesons(meson).getLambdaM();
    gslpp::complex Lambdamin = exp(-tmpq2 / MM / Lambdaplus) / Lambdaplus * (-gsl_sf_expint_Ei(tmpq2 / MM / Lambdaplus) + gslpp::complex::i() * M_PI);
 
    double u = up;
    return ((Tparminus(u, tmpq2)*6. * u * (1. - u)*
            (1 + threeGegen0 * (2. * u - 1)
            + threeGegen1otwo * ((10. * u - 5.)*(2. * u - 1.) - 1.))) / Lambdamin).real();
}

◆ Integrand_ReTparplus()

double MPll::Integrand_ReTparplus ( double up )

private

The real part of the integral involving $ T^{\parallel}_+ $ at fixed $ q^2 $, according to [Beneke:2001at] .

Parameters

[in] up dummy variable to be integrated out

Returns: $ Re T^{\parallel}_+ \Phi^{\parallel}$

Definition at line 1015 of file MPll.cpp.

{
    double u = up;
    return ((Tparplus(u, tmpq2)*6. * u * (1. - u)*
            (1 + threeGegen0 * (2. * u - 1)
            + threeGegen1otwo * ((10. * u - 5.)*(2. * u - 1.) - 1.))) / mySM.getMesons(meson).getLambdaM()).real();
}

◆ integrateDelta()

double MPll::integrateDelta	(	int	i,
		double	q_min,
		double	q_max
	)

The integral of $ \Delta_{i} $ from $q_{min}$ to $q_{max}$.

Parameters

[in]	i	index of the angular coefficient $ I_{i} $
[in]	q_min	minimum q^2 of the integral
[in]	q_max	maximum q^2 of the integral

Returns: $ <\Delta_{i}> $

Definition at line 1492 of file MPll.cpp.

{
    updateParameters();
 
    std::pair<double, double > qbin = std::make_pair(q_min, q_max);
 
    old_handler = gsl_set_error_handler_off();
 
    switch (i) {
        case 0:
            if (delta0Cached[qbin] == 0) {
                FD = convertToGslFunction(bind(&MPll::getDelta1c, &(*this), _1));
                if (gsl_integration_cquad(&FD, q_min, q_max, 1.e-2, 1.e-1, w_delta, &avaDelta, &errDelta, NULL) != 0) return std::numeric_limits<double>::quiet_NaN();
                cacheDelta0[qbin] = NN*avaDelta;
                delta0Cached[qbin] = 1;
            }
            return cacheDelta0[qbin];
            break;
        case 2:
            if (delta2Cached[qbin] == 0) {
                FD = convertToGslFunction(bind(&MPll::getDelta2c, &(*this), _1));
                if (gsl_integration_cquad(&FD, q_min, q_max, 1.e-2, 1.e-1, w_delta, &avaDelta, &errDelta, NULL) != 0) return std::numeric_limits<double>::quiet_NaN();
                cacheDelta2[qbin] = NN*avaDelta;
                delta2Cached[qbin] = 1;
            }
            return cacheDelta2[qbin];
            break;
        default:
            std::stringstream out;
            out << i;
            throw std::runtime_error("MPll::integrateDelta: index " + out.str() + " not implemented");
    }
 
    gsl_set_error_handler(old_handler);
 
}

◆ integrateSigma()

double MPll::integrateSigma	(	int	i,
		double	q_min,
		double	q_max
	)

The integral of $ \Sigma_{i} $ from $q_{min}$ to $q_{max}$.

Parameters

[in]	i	index of the angular coefficient $ I_{i} $
[in]	q_min	minimum q^2 of the integral
[in]	q_max	maximum q^2 of the integral

Returns: $ <\Sigma_{i}> $

Definition at line 1425 of file MPll.cpp.

{
    updateParameters();
    
    std::pair<double, double > qbin = std::make_pair(q_min, q_max);
 
    old_handler = gsl_set_error_handler_off();
 
    switch (i) {
        case 0:
            if (sigma0Cached[qbin] == 0) {
                FS = convertToGslFunction(bind(&MPll::getSigma1c, &(*this), _1));
                if (gsl_integration_cquad(&FS, q_min, q_max, 1.e-2, 1.e-1, w_sigma, &avaSigma, &errSigma, NULL) != 0) return std::numeric_limits<double>::quiet_NaN();
                cacheSigma0[qbin] = NN*avaSigma;
                sigma0Cached[qbin] = 1;
            }
            return cacheSigma0[qbin];
            break;
        case 2:
            if (sigma2Cached[qbin] == 0) {
                FS = convertToGslFunction(bind(&MPll::getSigma2c, &(*this), _1));
                if (gsl_integration_cquad(&FS, q_min, q_max, 1.e-2, 1.e-1, w_sigma, &avaSigma, &errSigma, NULL) != 0) return std::numeric_limits<double>::quiet_NaN();
                cacheSigma2[qbin] = NN*avaSigma;
                sigma2Cached[qbin] = 1;
            }
            return cacheSigma2[qbin];
            break;
        case 8:
            if (sigma8Cached[qbin] == 0) {
                FS = convertToGslFunction(bind(&MPll::getSigma6c, &(*this), _1));
                if (gsl_integration_cquad(&FS, q_min, q_max, 1.e-2, 1.e-1, w_sigma, &avaSigma, &errSigma, NULL) != 0) return std::numeric_limits<double>::quiet_NaN();
                cacheSigma8[qbin] = NN*avaSigma;
                sigma8Cached[qbin] = 1;
            }
            return cacheSigma8[qbin];
            break;
        default:
            std::stringstream out;
            out << i;
            throw std::runtime_error("MPll::integrateSigma: index " + out.str() + " not implemented");
    }
 
    gsl_set_error_handler(old_handler);
 
}

◆ integrateSigmaTree()

double MPll::integrateSigmaTree	(	double	q_min,
		double	q_max
	)

The integral of $ \Sigma_{tree} $ from $q_{min}$ to $q_{max}$ (arxiv/2301.06990)

Parameters

[in]	q_min	minimum q^2 of the integral
[in]	q_max	maximum q^2 of the integral

Returns: $ <\Sigma_{tree}> $

Definition at line 1529 of file MPll.cpp.

{
    if (lep != QCD::NEUTRINO_1 or meson != QCD::B_P or !NeutrinoTree_flag) return 0.;
 
    updateParameters();
    
    //phase space limit where tree-level contribution is relevant (0908.1174)
    double q_cut = (mtau2 - MP2) * (MM2 - mtau2) / mtau2;
    if (q_max >= q_cut) {
        if (q_min == 0.) return getintegratedSigmaTree();
        q_max = q_cut;
    }
    
    double prefactor = mySM.getMesons(meson).getLifetime() / HCUT * GF4 * VusVub_abs2 * fP2 * fM2 / (64. * M_PI2 * MM3 * Gammatau) * mtau2 * mtau;
    
    std::pair<double, double > qbin = std::make_pair(q_min, q_max);
 
    old_handler = gsl_set_error_handler_off();
 
    if (sigmaTreeCached[qbin] == 0) {
        FD = convertToGslFunction(bind(&MPll::SigmaTree, &(*this), _1));
        if (gsl_integration_cquad(&FD, q_min, q_max, 1.e-2, 1.e-1, w_sigmaTree, &avaSigmaTree, &errSigmaTree, NULL) != 0) return std::numeric_limits<double>::quiet_NaN();
        cacheSigmaTree[qbin] = avaSigmaTree;
        sigmaTreeCached[qbin] = 1;
    }
    return prefactor * cacheSigmaTree[qbin];
 
    gsl_set_error_handler(old_handler);
}

◆ reDC9fit()

double MPll::reDC9fit	(	double *	x,
		double *	p
	)

private

The fit function for the real part of the QCDF correction $ \Delta C_9^{\lambda} $.

Parameters

[in]	x	fit variable
[in]	p	fit parameters vector

Returns: $ f_{Re \Delta C_9^{\lambda}} $

Definition at line 1140 of file MPll.cpp.

{
    return p[0] / x[0] + p[1] + p[2] * x[0] + p[3] * x[0] * x[0] + p[4] * x[0] * x[0] * x[0] + p[5] * x[0] * x[0] * x[0] * x[0] + p[6] * x[0] * x[0] * x[0] * x[0] * x[0];
}

◆ S_L()

double MPll::S_L ( double q2 )

The helicity form factor $ S_L $.

Parameters

[in] q2 $q^2$ of the decay

Returns: $ S_L $

Definition at line 964 of file MPll.cpp.

{
    return S_L_pre * f_0(q2);
}

◆ SigmaTree()

double MPll::SigmaTree ( double q2 )

private

Definition at line 1559 of file MPll.cpp.

{
    return MM2 * (mtau2 - MP2) - mtau2 * (mtau2 + q2 - MP2);
}

◆ spline_QCDF_func()

void MPll::spline_QCDF_func ( )

private

Definition at line 1248 of file MPll.cpp.

{
    int dim_DC = GSL_INTERP_DIM_DC;
    double min = 0.001;
    double interval_DC = (9.9 - min) / ((double) dim_DC);
    double q2_spline_DC[dim_DC];
    double fq2_Re_deltaC7_QCDF[dim_DC], fq2_Im_deltaC7_QCDF[dim_DC], fq2_Re_deltaC9_QCDF[dim_DC], fq2_Im_deltaC9_QCDF[dim_DC];
 
    for (int i = 0; i < dim_DC; i++) {
        q2_spline_DC[i] = min + (double) i*interval_DC;
        fq2_Re_deltaC7_QCDF[i] = deltaC7_QCDF(q2_spline_DC[i], false).real();
        fq2_Im_deltaC7_QCDF[i] = deltaC7_QCDF(q2_spline_DC[i], false).imag();
        fq2_Re_deltaC9_QCDF[i] = deltaC9_QCDF(q2_spline_DC[i], false).real();
        fq2_Im_deltaC9_QCDF[i] = deltaC9_QCDF(q2_spline_DC[i], false).imag();
 
    }
 
    gsl_spline_init(spline_Re_deltaC7_QCDF, q2_spline_DC, fq2_Re_deltaC7_QCDF, dim_DC);
    gsl_spline_init(spline_Im_deltaC7_QCDF, q2_spline_DC, fq2_Im_deltaC7_QCDF, dim_DC);
    gsl_spline_init(spline_Re_deltaC9_QCDF, q2_spline_DC, fq2_Re_deltaC9_QCDF, dim_DC);
    gsl_spline_init(spline_Im_deltaC9_QCDF, q2_spline_DC, fq2_Im_deltaC9_QCDF, dim_DC);
 
 
}

◆ T_L()

gslpp::complex MPll::T_L ( double q2 )

The helicity form factor $ T_L^0 $.

Parameters

[in] q2 $q^2$ of the decay

Returns: $ T_L^{\lambda} $

Definition at line 959 of file MPll.cpp.

{
    return /*gslpp::complex::i() */ sqrt(lambda(q2) * q2) / (twoMM2_MMpMP) * f_T(q2);
}

◆ Tparminus()

gslpp::complex MPll::Tparminus	(	double	u,
		double	q2
	)

private

The $ T^{\parallel}_- $ function from [Khodjamirian:2012rm] .

Parameters

[in]	u	dummy variable to be integrated out
[in]	q2	$q^2$ of the decay

Returns: $ T^{\parallel}_- $

Definition at line 1008 of file MPll.cpp.

{
    double ubar = 1. - u;
    return -spectator_charge * (8. * C_8Lh / (ubar + u * q2 / MM2)
            + sixMMoMb * H_c(ubar * MM2 + u * q2, mu_h * mu_h) * C_2Lh_bar);
}

◆ Tparplus()

gslpp::complex MPll::Tparplus	(	double	u,
		double	q2
	)

private

The $ T^{\parallel}_+ $ function from [Khodjamirian:2012rm] .

Parameters

[in]	u	dummy variable to be integrated out
[in]	q2	$q^2$ of the decay

Returns: $ T^{\parallel}_+ $

Definition at line 995 of file MPll.cpp.

{
    Ee = (MM2 - q2) / twoMM;
    ubar = 1. - u;
    arg1 = (fourMc2 - gslpp::complex::i()*1.e-10) / (ubar * MM2 + u * q2) - 1.;
    B01 = -2. * sqrt(arg1) * arctan(1. / sqrt(arg1));
    arg1 = (fourMc2 - gslpp::complex::i()*1.e-10) / q2 - 1.;
    B00 = -2. * sqrt(arg1) * arctan(1. / sqrt(arg1));
 
    gslpp::complex tpar = twoMM / Ee / ubar * I1(u, q2) + (ubar * MM2 + u * q2) / Ee / Ee / ubar / ubar * (B01 - B00);
    return -MM / Mb * mySM.getQuarks(QCD::CHARM).getCharge() * tpar*C_2Lh_bar;
}

◆ V_L()

gslpp::complex MPll::V_L ( double q2 )

The helicity form factor $ V_L^0 $.

Parameters

[in] q2 $q^2$ of the decay

Returns: $ V_L^{\lambda} $

Definition at line 954 of file MPll.cpp.

{
    return /*gslpp::complex::i() */ sqrt(lambda(q2)) / (twoMM * sqrt(q2)) * f_plus(q2);
}

Member Data Documentation

◆ ale

double MPll::ale

private

alpha electromagnetic

Definition at line 331 of file MPll.h.

◆ alpha_s_mub

double MPll::alpha_s_mub

private

@f\aplha_s(\mu_b)$ \( */ double angmomP; /**<angular momentum of meson P; for a resonance, it's replaced by its spin */ int etaP; /**<parity of meson P */ double MW; /**<W boson mass */ gslpp::complex lambda_t; /**<Vckm factor */ gslpp::complex h_0; /**<parameter that contains the contribution from the hadronic hamiltonian */ gslpp::complex h_1; /**<parameter that contains the contribution from the hadronic hamiltonian */ gslpp::complex h_2; /**<parameter that contains the contribution from the hadronic hamiltonian */ gslpp::complex r_1; gslpp::complex r_2; gslpp::complex Delta_C9; gslpp::complex exp_Phase; double Delta_C7_U; double Delta_C9_U; /*LATTICE fit parameters*/ double b_0_fplus;/**<LATTICE fit parameter */ double b_1_fplus;/**<LATTICE fit parameter */ double b_2_fplus;/**<LATTICE fit parameter */ double m_fit2_fplus_lat;/**<LATTICE fit parameter */ double b_0_fT;/**<LATTICE fit parameter */ double b_1_fT;/**<LATTICE fit parameter */ double b_2_fT;/**<LATTICE fit parameter */ double m_fit2_fT_lat;/**<LATTICE fit parameter */ double b_0_f0;/**<LATTICE fit parameter */ double b_1_f0;/**<LATTICE fit parameter */ double b_2_f0;/**<LATTICE fit parameter */ double m_fit2_f0_lat;/**<LATTICE fit parameter */ /*LCSR fit parameters*/ double r_1_fplus;/**<LCSR fit parameter */ double r_2_fplus;/**<LCSR fit parameter */ double m_fit2_fplus;/**<LCSR fit parameter */ double r_1_fT;/**<LCSR fit parameter */ double r_2_fT;/**<LCSR fit parameter */ double m_fit2_fT;/**<LCSR fit parameter */ double r_2_f0;/**<LCSR fit parameter */ double m_fit2_f0;/**<LCSR fit parameter */ gslpp::vector<gslpp::complex> ** allcoeff;/**<vector that contains the Wilson coeffients */ gslpp::vector<gslpp::complex> ** allcoeffh;/**<Vector that contains the Wilson coeffients at scale \)\mu_h $ */ gslpp::vector<gslpp::complex> ** allcoeffprime;/**<vector that contains the primed Wilson coeffients */ gslpp::vector<gslpp::complex> ** allcoeff_noSM;/**<vector that contains the Wilson coeffients without the SM contributions.*/ gslpp::vector<gslpp::complex> ** allcoeff_nu;/**<Vector that contains the Wilson coeffients */ gslpp::vector<gslpp::complex> ** allcoeff_noSM_nu;/**<Vector that contains the Wilson coeffients without the SM contribution.*/ gslpp::complex C_1;/**<Wilson coeffients $@_fakenlC_1 $*/ gslpp::complex C_1L_bar;/**<Wilson coeffients $@_fakenlC_1 $*/ gslpp::complex C_1Lh_bar;/**<Wilson coeffients $@_fakenlC_1 $*/ gslpp::complex C_2;/**<Wilson coeffients $@_fakenlC_2 $*/ gslpp::complex C_2L_bar;/**<Leading order Wilson coeffients $@_fakenlC_2 $*/ gslpp::complex C_2Lh_bar;/**<Leading order Wilson coeffients $@_fakenlC_2 $ at scale $\mu_h $*/ gslpp::complex C_3;/**<Wilson coeffients $@_fakenlC_3 $*/ gslpp::complex C_4;/**<Wilson coeffients $@_fakenlC_4 $*/ gslpp::complex C_5;/**<Wilson coeffients $@_fakenlC_5 $*/ gslpp::complex C_6;/**<Wilson coeffients $@_fakenlC_6 $*/ gslpp::complex C_7;/**<Wilson coeffients $@_fakenlC_7 $*/ gslpp::complex C_8;/**<Wilson coeffients $@_fakenlC_8 $*/ gslpp::complex C_8L;/**<Leading order Wilson coeffients $@_fakenlC_8 $*/ gslpp::complex C_8Lh;/**<Leading order Wilson coeffients $@_fakenlC_8 $ at scale $\mu_h $*/ gslpp::complex C_9;/**<Wilson coeffients $@_fakenlC_9 $*/ gslpp::complex C_10;/**<Wilson coeffients $@_fakenlC_{10} $*/ gslpp::complex C_S;/**<Wilson coeffients $@_fakenlC_S $*/ gslpp::complex C_P;/**<Wilson coeffients $@_fakenlC_P $*/ gslpp::complex C_7p;/**<Wilson coeffients $@_fakenlC_7' $*/ gslpp::complex C_9p;/**<Wilson coeffients $@_fakenlC_9' $*/ gslpp::complex C_10p;/**<Wilson coeffients $@_fakenlC_{10}' $*/ gslpp::complex C_Sp;/**<Wilson coeffients $@_fakenlC_S' $*/ gslpp::complex C_Pp;/**<Wilson coeffients $@_fakenlC_P' $*/ gslpp::complex C_L_nunu_e;/**<Wilson coeffients $@_fakenlC_L^{\nu_1\nu_1}' $*/ gslpp::complex C_R_nunu_e;/**<Wilson coeffients $@_fakenlC_R^{\nu_1\nu_1}' $*/ gslpp::complex C_L_nunu_mu;/**<Wilson coeffients $@_fakenlC_L^{\nu_2\nu_2}' $*/ gslpp::complex C_R_nunu_mu;/**<Wilson coeffients $@_fakenlC_R^{\nu_2\nu_2}' $*/ gslpp::complex C_L_nunu_tau;/**<Wilson coeffients $@_fakenlC_L^{\nu_3\nu_3}' $*/ gslpp::complex C_R_nunu_tau;/**<Wilson coeffients $@_fakenlC_R^{\nu_3\nu_3}' $*/ gsl_interp_accel *acc_Re_deltaC7_QCDF; gsl_interp_accel *acc_Im_deltaC7_QCDF; gsl_interp_accel *acc_Re_deltaC9_QCDF; gsl_interp_accel *acc_Im_deltaC9_QCDF; gsl_spline *spline_Re_deltaC7_QCDF; gsl_spline *spline_Im_deltaC7_QCDF; gsl_spline *spline_Re_deltaC9_QCDF; gsl_spline *spline_Im_deltaC9_QCDF; std::vector<double> ReDeltaC9;/**<Vector that samples the QCDF $\Delta C_9 $ */ std::vector<double> ImDeltaC9;/**<Vector that samples the QCDF $\Delta C_9 $ */ std::vector<double> myq2;/**<Variable used to compute the QCDF $\Delta C_9 $ */ TFitResultPtr refres;/**<Vector that contains the fitted QCDF $\Delta C_9 $ */ TFitResultPtr imfres;/**<Vector that contains the fitted QCDF $\Delta C_9 $ */ TGraph gr1;/**<Tgraph to be used for fitting the QCDF $\Delta C_9 $ */ TGraph gr2;/**<Tgraph to be used for fitting the QCDF $\Delta C_9 $ */ TF1 reffit;/**<TF1 to be used for fitting the QCDF $\Delta C_9 $ */ TF1 imffit;/**<TF1 to be used for fitting the QCDF $\Delta C_9 $ */ double tmpq2;/**<Variable used to compute the QCDF $\Delta C_9 \( */ gslpp::complex H_0_pre;/**< Cache variable */ gslpp::complex H_0_WC;/**< Cache variable */ gslpp::complex H_c_WC;/**< Cache variable */ gslpp::complex H_b_WC;/**< Cache variable */ gslpp::complex ihalfMPI;/**< Cache variable */ double fournineth;/**< Cache variable */ double half;/**< Cache variable */ double twothird;/**< Cache variable */ double Mc2;/**< Cache variable */ double Mb2;/**< Cache variable */ double logMc;/**< Cache variable */ double logMb;/**< Cache variable */ double mu_b2;/**< Cache variable */ double fourMc2;/**< Cache variable */ double fourMb2;/**< Cache variable */ double Mlep2;/**< Cache variable */ double NN;/**< Cache variable */ double MM2;/**< Cache variable */ double MM4;/**< Cache variable */ double MP2;/**< Cache variable */ double MP4;/**< Cache variable */ double MM2mMP2;/**< Cache variable */ double twoMP2;/**< Cache variable */ double twoMM;/**< Cache variable */ double twoMM2;/**< Cache variable */ double twoMM2_MMpMP;/**< Cache variable */ double twoMM_MbpMs;/**< Cache variable */ double S_L_pre;/**< Cache variable */ double fourMM2;/**< Cache variable */ double twoMboMM;/**< Cache variable */ double sixteenM_PI2;/**< Cache variable */ double ninetysixM_PI3MM3;/**< Cache variable */ double MboMW;/**< Cache variable */ double MboMM;/**< Cache variable */ double MsoMb;/**< Cache variable */ double twoMlepMb;/**< Cache variable */ double threeGegen0;/**< Cache variable */ double threeGegen1otwo;/**< Cache variable */ double M_PI2osix;/**< Cache variable */ double twoMc2;/**< Cache variable */ double sixMMoMb;/**< Cache variable */ double CF;/**< Cache variable */ double deltaT_0;/**< Cache variable */ double deltaT_1par;/**< Cache variable */ gslpp::complex ubar;/**<Variable used to compute the QCDF \)\Delta C_9 $ */ gslpp::complex arg1;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex B01;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex B00;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex xp;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex xm;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex yp;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex ym;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex L1xp;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex L1xm;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex L1yp;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex L1ym;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F87_0;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F87_1;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F87_2;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F87_3;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F29_0;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F29_L1;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F29_1;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F29_2;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F29_3;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F29_L1_1;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F29_L1_2;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F29_L1_3;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F27_0;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F27_1;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F27_2;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F27_3;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F27_L1_1;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F27_L1_2;/**<Variable used to compute the QCDF $\Delta C_9 $ */ gslpp::complex F27_L1_3;/**<Variable used to compute the QCDF $\Delta C_9 $ */ double F89_0;/**<Variable used to compute the QCDF $\Delta C_9 $ */ double F89_1;/**<Variable used to compute the QCDF $\Delta C_9 $ */ double F89_2;/**<Variable used to compute the QCDF $\Delta C_9 $ */ double F89_3;/**<Variable used to compute the QCDF $\Delta C_9 $ */ double Ee;/**<Variable used to compute the QCDF $\Delta C_9 \( */ //additional variables for B to K nu nu const double M_PI2 = M_PI * M_PI; double GF4; double MM3; double fM2; double fP2; double mtau; double mtau2; double Gammatau; double VusVub_abs2; unsigned int fplus_lat_updated;/**< Cache variable */ gslpp::vector<double> fplus_lat_cache;/**< Cache variable */ unsigned int fT_lat_updated;/**< Cache variable */ gslpp::vector<double> fT_lat_cache;/**< Cache variable */ unsigned int f0_lat_updated;/**< Cache variable */ gslpp::vector<double> f0_lat_cache;/**< Cache variable */ unsigned int fplus_updated;/**< Cache variable */ gslpp::vector<double> fplus_cache;/**< Cache variable */ unsigned int fT_updated;/**< Cache variable */ gslpp::vector<double> fT_cache;/**< Cache variable */ unsigned int f0_updated;/**< Cache variable */ double f0_cache;/**< Cache variable */ unsigned int k2_updated;/**< Cache variable */ gslpp::vector<double> k2_cache;/**< Cache variable */ unsigned int beta_updated;/**< Cache variable */ double beta_cache;/**< Cache variable */ unsigned int lambda_updated;/**< Cache variable */ unsigned int F_updated;/**< Cache variable */ unsigned int VL_updated;/**< Cache variable */ unsigned int TL_updated;/**< Cache variable */ unsigned int SL_updated;/**< Cache variable */ gslpp::vector<double> SL_cache;/**< Cache variable */ unsigned int N_updated;/**< Cache variable */ gslpp::vector<double> N_cache;/**< Cache variable */ gslpp::complex Nc_cache;/**< Cache variable */ unsigned int C_1_updated;/**< Cache variable */ gslpp::complex C_1_cache;/**< Cache variable */ unsigned int C_2_updated;/**< Cache variable */ gslpp::complex C_2_cache;/**< Cache variable */ unsigned int C_3_updated;/**< Cache variable */ gslpp::complex C_3_cache;/**< Cache variable */ unsigned int C_4_updated;/**< Cache variable */ gslpp::complex C_4_cache;/**< Cache variable */ unsigned int C_5_updated;/**< Cache variable */ gslpp::complex C_5_cache;/**< Cache variable */ unsigned int C_6_updated;/**< Cache variable */ gslpp::complex C_6_cache;/**< Cache variable */ unsigned int C_7_updated;/**< Cache variable */ gslpp::complex C_7_cache;/**< Cache variable */ unsigned int C_9_updated;/**< Cache variable */ gslpp::complex C_9_cache;/**< Cache variable */ unsigned int C_10_updated;/**< Cache variable */ gslpp::complex C_10_cache;/**< Cache variable */ unsigned int C_7p_updated;/**< Cache variable */ gslpp::complex C_7p_cache;/**< Cache variable */ unsigned int C_9p_updated;/**< Cache variable */ gslpp::complex C_9p_cache;/**< Cache variable */ unsigned int C_10p_updated;/**< Cache variable */ gslpp::complex C_10p_cache;/**< Cache variable */ unsigned int C_S_updated;/**< Cache variable */ gslpp::complex C_S_cache;/**< Cache variable */ unsigned int C_P_updated;/**< Cache variable */ gslpp::complex C_P_cache;/**< Cache variable */ unsigned int C_Sp_updated;/**< Cache variable */ gslpp::complex C_Sp_cache;/**< Cache variable */ unsigned int C_Pp_updated;/**< Cache variable */ gslpp::complex C_Pp_cache;/**< Cache variable */ unsigned int C_2Lh_updated;/**< Cache variable */ gslpp::complex C_2Lh_cache;/**< Cache variable */ unsigned int C_8Lh_updated;/**< Cache variable */ gslpp::complex C_8Lh_cache;/**< Cache variable */ unsigned int C_L_nunu_e_updated;/**< Cache variable */ unsigned int C_L_nunu_mu_updated;/**< Cache variable */ unsigned int C_L_nunu_tau_updated;/**< Cache variable */ gslpp::complex C_L_nunu_e_cache;/**< Cache variable */ gslpp::complex C_L_nunu_mu_cache;/**< Cache variable */ gslpp::complex C_L_nunu_tau_cache;/**< Cache variable */ unsigned int C_R_nunu_e_updated;/**< Cache variable */ unsigned int C_R_nunu_mu_updated;/**< Cache variable */ unsigned int C_R_nunu_tau_updated;/**< Cache variable */ gslpp::complex C_R_nunu_e_cache;/**< Cache variable */ gslpp::complex C_R_nunu_mu_cache;/**< Cache variable */ gslpp::complex C_R_nunu_tau_cache;/**< Cache variable */ unsigned int Yupdated;/**< Cache variable */ gslpp::vector<double> Ycache;/**< Cache variable */ unsigned int H_V0updated;/**< Cache variable */ gslpp::vector<double> H_V0cache;/**< Cache variable */ gslpp::complex H_V0Ccache[3];/**< Cache variable */ gslpp::complex H_V0Ccache_dispersion[4];/**< Cache variable */ unsigned int H_A0updated;/**< Cache variable */ unsigned int H_Supdated;/**< Cache variable */ gslpp::vector<double> H_Scache;/**< Cache variable */ unsigned int H_P_updated;/**< Cache variable */ gslpp::vector<double> H_P_cache;/**< Cache variable */ unsigned int I0_updated;/**< Cache variable */ unsigned int I2_updated;/**< Cache variable */ unsigned int I8_updated;/**< Cache variable */ unsigned int Itree_updated;/**< Cache variable */ gslpp::vector<double> Itree_cache;/**< Cache variable */ std::map<std::pair<double, double>, unsigned int > I1Cached;/**< Cache variable */ std::map<std::pair<double, double>, unsigned int > sigma0Cached;/**< Cache variable */ std::map<std::pair<double, double>, unsigned int > sigma2Cached;/**< Cache variable */ std::map<std::pair<double, double>, unsigned int > sigma8Cached;/**< Cache variable */ std::map<std::pair<double, double>, unsigned int > delta0Cached;/**< Cache variable */ std::map<std::pair<double, double>, unsigned int > delta2Cached;/**< Cache variable */ std::map<std::pair<double, double>, unsigned int > sigmaTreeCached;/**< Cache variable */ double avaSigma;/**< Gsl integral variable */ double avaDelta;/**< Gsl integral variable */ double avaSigmaTree;/**< Gsl integral variable */ double avaDTPPR;/**< Gsl integral variable */ double errSigma;/**< Gsl integral variable */ double errDelta;/**< Gsl integral variable */ double errSigmaTree;/**< Gsl integral variable */ double errDTPPR;/**< Gsl integral variable */ gsl_function FS;/**< Gsl integral variable */ gsl_function FD;/**< Gsl integral variable */ gsl_function DTPPR;/**< Gsl integral variable */ gsl_integration_cquad_workspace * w_sigma;/**< Gsl integral variable */ gsl_integration_cquad_workspace * w_delta;/**< Gsl integral variable */ gsl_integration_cquad_workspace * w_sigmaTree;/**< Gsl integral variable */ gsl_integration_cquad_workspace * w_DTPPR;/**< Gsl integral variable */ gsl_error_handler_t * old_handler; /**< GSL error handler store */ std::map<std::pair<double, double>, gslpp::complex > cacheI1;/**< Cache variable */ std::map<std::pair<double, double>, double > cacheSigma0;/**< Gsl integral variable */ std::map<std::pair<double, double>, double > cacheSigma2;/**< Gsl integral variable */ std::map<std::pair<double, double>, double > cacheSigma8;/**< Gsl integral variable */ std::map<std::pair<double, double>, double > cacheDelta0;/**< Gsl integral variable */ std::map<std::pair<double, double>, double > cacheDelta2;/**< Gsl integral variable */ std::map<std::pair<double, double>, double > cacheSigmaTree;/**< Gsl integral variable */ std::map<double, unsigned int> deltaTparpCached;/**< Cache variable */ std::map<double, unsigned int> deltaTparmCached;/**< Cache variable */ std::map<double, gslpp::complex> cacheDeltaTparp;/**< Cache variable */ std::map<double, gslpp::complex> cacheDeltaTparm;/**< Cache variable */ unsigned int deltaTparpupdated;/**< Cache variable */ unsigned int deltaTparmupdated;/**< Cache variable */ unsigned int T_updated;/**< Cache variable */ gslpp::vector<double> T_cache;/**< Cache variable */ /** @brief The update parameter method for MPll. */ void updateParameters(); /** @brief The caching method for MPll. */ void checkCache(); /** @brief The first fit function from arXiv:hep-ph/0412079v1,\) f_1^{LATTICE} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] r_1 fit parameter @param[in] r_2 fit parameter @param[in] m_fit2 fit parameter @return $ f_1^{LATTICE} $ */ double LCSR_fit1(double q2, double r_1, double r_2, double m_fit2); /** @brief The second fit function from arXiv:hep-ph/0412079v1, $ f_2^{LATTICE} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] r_2 fit parameter @param[in] m_fit2 fit parameter @return $ f_2^{LATTICE} $ */ double LCSR_fit2(double q2, double r_2, double m_fit2); /** @brief The fit function from @cite Straub:2015ica, $ FF^{\rm fit} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] b_0 fit parameter @param[in] b_1 fit parameter @param[in] b_2 fit parameter @param[in] m_fit2 square of the nearest resonance mass @return $ FF^{\rm fit} $ */ double LCSR_fit3(double q2, double b_0, double b_1, double b_2, double m_fit2); /** @brief The z function from arXiv:hep-ph/0412079v1, $ z $. @param[in] q2 $@_fakenlq^2 $ of the decay @return $ z $ */ double zeta(double q2); /** @brief The z function used for the DM Form Factors, $ z $. @param[in] q2 $@_fakenlq^2 $ of the decay @return $ z $ */ double zeta_DM(double q2); /** @brief The first fit function from arXiv:hep-ph/0412079v1,$ f_+^{DM} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] m_fit2 fit parameter @return $ f_+^{DM} $ */ double phiplus_DM(double q2, double m_fit2); /** @brief The first fit function from arXiv:hep-ph/0412079v1,$ f_+^{DM} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] m_fit2 fit parameter @return $ f_+^{DM} $ */ double phi0_DM(double q2); /** @brief The first fit function from arXiv:hep-ph/0412079v1,$ f_+^{DM} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] m_fit2 fit parameter @return $ f_+^{DM} $ */ double phiT_DM(double q2, double m_fit2); /** @brief The first fit function from arXiv:hep-ph/0412079v1,$ f_+^{DM} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] b_0 fit parameter @param[in] b_1 fit parameter @param[in] b_2 fit parameter @param[in] m_fit2 fit parameter @return $ f_+^{DM} $ */ double fplus_DM(double q2, double b_0, double b_1, double b_2, double m_fit2); /** @brief The first fit function from arXiv:hep-ph/0412079v1,$ f_0^{DM} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] b_0 fit parameter @param[in] b_1 fit parameter @param[in] b_2 fit parameter @param[in] m_fit2 fit parameter @return $ f_0^{DM} $ */ double f0_DM(double q2, double b_0, double b_1, double b_2); /** @brief The first fit function from arXiv:hep-ph/0412079v1,$ f_T^{DM} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] b_0 fit parameter @param[in] b_1 fit parameter @param[in] b_2 fit parameter @param[in] m_fit2 fit parameter @return $ f_T^{DM} $ */ double fT_DM(double q2, double b_0, double b_1, double b_2, double m_fit2); /** @brief The first fit function from arXiv:1509.06235v2,$ f_1^{LATTICE} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] b_0 fit parameter @param[in] b_1 fit parameter @param[in] b_2 fit parameter @param[in] m_fit2 fit parameter @return $ f_1^{LATTICE} $ */ double LATTICE_fit1(double q2, double b_0, double b_1, double b_2, double m_fit2); /** @brief The second fit function from arXiv:1509.06235v2, $ f_2^{LATTICE} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] b_0 fit parameter @param[in] b_1 fit parameter @param[in] b_2 fit parameter @param[in] m_fit2 fit parameter @return $ f_2^{LATTICE} $ */ double LATTICE_fit2(double q2, double b_0, double b_1, double b_2, double m_fit2); /** @brief The form factor $ f_+ $. @param[in] q2 $@_fakenlq^2 $ of the decay @return $ f_+ $ */ double f_plus(double q2); /** @brief The form factor $ f_T $. @param[in] q2 $@_fakenlq^2 $ of the decay @return $ f_T $ */ double f_T(double q2); /** @brief The form factor $ f_0 $. @param[in] q2 $@_fakenlq^2 $ of the decay @return $ f_0 $ */ double f_0(double q2); /** @brief The $ h(q^2,0) $ function involved into $ C_9^{eff} $. @param[in] q2 $@_fakenlq^2 $ of the decay @return $ h(q^2,0) $ */ gslpp::complex H_0(double q2); /** @brief The $ h(q^2,m_c) $ function involved into $ C_9^{eff} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] mu2 mass scale @return $ h(q^2,m_c) $ */ gslpp::complex H_c(double q2, double mu2); /** @brief The $ h(q^2,m_b) $ function involved into $ C_9^{eff} $. @param[in] q2 $@_fakenlq^2 $ of the decay @param[in] mu2 mass scale @return $ h(q^2,m_b) $ */ gslpp::complex H_b(double q2, double mu2); /** @brief The $ Y(q^2) $ function involved into $ C_9^{eff} $. @param[in] q2 $@_fakenlq^2 $ of the decay @return $ Y(q^2) $ */ gslpp::complex Y(double q2); /** @brief The square of the 3-momentum of the recoiling meson in the M rest frame, $ k^2 $ . @param[in] q2 $@_fakenlq^2 $ of the decay @return $ k^2 $ */ double k2 (double q2); /** @brief The factor $ \beta $ used in the angular coefficients $I_i $. @param[in] q2 $@_fakenlq^2 $ of the decay @return $ \beta $ */ double beta (double q2); /** @brief The factor $ \beta^2 $ used in the angular coefficients $I_i $. @param[in] q2 $@_fakenlq^2 $ of the decay @return $ \beta^2 $ */ double beta2 (double q2); /** @brief The factor $ \lambda $ used in the angular coefficients $I_i $. @param[in] q2 $@_fakenlq^2 $ of the decay @return $ \lambda $ */ double lambda(double q2); /** @brief The factor $ F $ used in the angular coefficients $I_i $. @param[in] q2 $@_fakenlq^2 $ of the decay @return $ F $ */ double F(double q2); /** @brief The angular coefficient $ I_{1c} $ . @param[in] q2 $@_fakenlq^2 $ of the decay @return $ I_{1c} $ */ double I_1c(double q2); /** @brief The angular coefficient $ I_{2c} $ . @param[in] q2 $@_fakenlq^2 $ of the decay @return $ I_{2c} $ */ double I_2c(double q2); /** @brief The angular coefficient $ I_{6c} $ . @param[in] q2 $@_fakenlq^2 $ of the decay @return $ I_{6c} $ */ double I_6c(double q2); /** @brief The CP asymmetry $ \Delta_{i} $ . @param[in] i index of the angular coefficient $ I_{i} $ @param[in] q2 $@_fakenlq^2 $ of the decay @return $ \Delta_{i} $ */ double Delta(int i, double q2); /** @brief The value of $ \Sigma_{tree}: contains the full q2-dependence but neglects a "prefactor"

Parameters

[in] q2 $q^2$ of the decay

Returns: $ <\Sigma{tree}> $

Definition at line 344 of file MPll.h.

◆ dispersion

bool MPll::dispersion = false

private

Definition at line 322 of file MPll.h.

◆ FixedWCbtos

bool MPll::FixedWCbtos = false

private

Definition at line 323 of file MPll.h.

◆ GF

double MPll::GF

private

Fermi constant

Definition at line 330 of file MPll.h.

◆ lep

QCD::lepton MPll::lep

private

Final leptons type

Definition at line 316 of file MPll.h.

◆ Mb

double MPll::Mb

private

b quark mass

Definition at line 335 of file MPll.h.

◆ mb_pole

double MPll::mb_pole

private

b quark pole mass

Definition at line 339 of file MPll.h.

◆ Mc

double MPll::Mc

private

c quark mass

Definition at line 338 of file MPll.h.

◆ mc_pole

double MPll::mc_pole

private

c quark pole mass

Definition at line 340 of file MPll.h.

◆ meson

QCD::meson MPll::meson

private

Initial meson type

Definition at line 317 of file MPll.h.

◆ mJ2

double MPll::mJ2

private

Definition at line 328 of file MPll.h.

◆ Mlep

double MPll::Mlep

private

muon mass

Definition at line 332 of file MPll.h.

◆ MM

double MPll::MM

private

initial meson mass

Definition at line 333 of file MPll.h.

◆ MP

double MPll::MP

private

final pseudoscalar meson mass

Definition at line 334 of file MPll.h.

◆ MPll_DM_flag

bool MPll::MPll_DM_flag

private

A flag for switching to DM FF parameterization

Definition at line 326 of file MPll.h.

◆ MPll_GRvDV_flag

bool MPll::MPll_GRvDV_flag

private

A flag for switching to GRvDV parameterization

Definition at line 325 of file MPll.h.

◆ MPll_Lattice_flag

bool MPll::MPll_Lattice_flag

private

A flag for switching to LATTICE FF parameterization

Definition at line 324 of file MPll.h.

◆ mpllParameters

std::vector<std::string> MPll::mpllParameters

private

The string of mandatory MPll parameters

Definition at line 319 of file MPll.h.

◆ Ms

double MPll::Ms

private

s quark mass

Definition at line 341 of file MPll.h.

◆ mu_b

double MPll::mu_b

private

b mass scale

Definition at line 336 of file MPll.h.

◆ mu_h

double MPll::mu_h

private

$\sqrt{\mu_b*\lambda_{QCD}}$

Definition at line 337 of file MPll.h.

◆ myF_1

std::unique_ptr<F_1> MPll::myF_1

private

Definition at line 320 of file MPll.h.

◆ myF_2

std::unique_ptr<F_2> MPll::myF_2

private

Definition at line 321 of file MPll.h.

◆ mySM

const StandardModel& MPll::mySM

private

Model type

Definition at line 315 of file MPll.h.

◆ NeutrinoTree_flag

bool MPll::NeutrinoTree_flag

private

Definition at line 327 of file MPll.h.

◆ pseudoscalar

QCD::meson MPll::pseudoscalar

private

Final pseudoscalar meson type

Definition at line 318 of file MPll.h.

◆ spectator_charge

double MPll::spectator_charge

private

spectator quark charge

Definition at line 342 of file MPll.h.

◆ width

double MPll::width

private

initial meson width

Definition at line 343 of file MPll.h.

The documentation for this class was generated from the following files:

Label	LaTeX symbol	Description
r_1_fplus, r_2_fplus, m_fit2_fplus	\(r_1^{f_+}, r_2^{f_+}, m_{fit}^{2,f_+}\)	The fit parameters for the LCSR form factor \(f_+\) of the \(B\to K\).
r_1_fT, r_2_fT, m_fit2_fT	\(r_1^{f_T}, r_2^{f_T}, m_{fit}^{2,f_T}\)	The fit parameters for the LCSR form factor \(f_T\) of the \(B\to K\).
r_2_f0, m_fit2_f0	\(r_2^{f_0}, m_{fit}^{2,f_0}\)	The fit parameters for the LCSR form factor \(f_0\) of the \(B\to K\).
absh_0_MP, absh_0_1_MP	\(\mathrm{Abs}h_0^{(0)}, \mathrm{Abs}h_0^{(1)}\)	The constant and linear terms of the absolute value of the hadronic parameter \(h_0\) of the \(B\to K\).
argh_0_MP, argh_0_1_MP	\(\mathrm{Arg}h_0^{(0)}, \mathrm{Arg}h_0^{(1)}\)	The constant and linear terms of the argument of the hadronic parameter \(h_0\) of the \(B\to K\).

[in]	i	index of the angular coefficient \( I_{i} \)
[in]	q_min	minimum q^2 of the integral
[in]	q_max	maximum q^2 of the integral

Detailed Description

MPll parameters

Public Member Functions

Private Member Functions

Private Attributes

Constructor & Destructor Documentation

◆ MPll()

◆ ~MPll()

Member Function Documentation

◆ Cpar()

◆ deltaC7_QCDF()

◆ DeltaC9()

◆ DeltaC9_KD()

◆ deltaC9_QCDF()

◆ deltaTpar()

◆ F19()

◆ F27()

◆ F29()

◆ F87()

◆ F89()

◆ fDeltaC9()

◆ fit_DeltaC9_mumu()

◆ funct_g()

◆ getDelta1c()

◆ getDelta2c()

◆ getintegratedSigmaTree()

◆ getSigma()

◆ getSigma1c()

◆ getSigma2c()

◆ getSigma6c()

◆ getwidth()

◆ H_A()

◆ h_lambda()

◆ H_nunu()

◆ H_P()

◆ H_S()

◆ H_V()

◆ I1()

◆ imDC9fit()

◆ initializeMPllParameters()

◆ Integrand_ImTpar_pm()

◆ Integrand_ImTparminus()

◆ Integrand_ImTparplus()

◆ Integrand_ReTpar_pm()

◆ Integrand_ReTparminus()

◆ Integrand_ReTparplus()

◆ integrateDelta()

◆ integrateSigma()

◆ integrateSigmaTree()

◆ reDC9fit()

◆ S_L()

◆ SigmaTree()

◆ spline_QCDF_func()

◆ T_L()

◆ Tparminus()

◆ Tparplus()

◆ V_L()

Member Data Documentation

◆ ale

◆ alpha_s_mub

◆ dispersion

◆ FixedWCbtos

◆ GF

◆ lep

◆ Mb

◆ mb_pole

◆ Mc

◆ mc_pole

◆ meson

◆ mJ2

◆ Mlep

◆ MM

◆ MP

◆ MPll_DM_flag

◆ MPll_GRvDV_flag

◆ MPll_Lattice_flag

◆ mpllParameters

◆ Ms

◆ mu_b

◆ mu_h